// An AA tree in computer science is a red-black tree with 
// one additional rule. 
// Unlike red-black trees, RED nodes on an AA tree can only 
// be added as a right subchild. 
// In other words, no RED node can be a left subchild. 
// This results in the simulation of a 2-3 tree instead of a 2-3-4 tree, 
// which greatly simplifies the maintenance operations.

// The following code shows how to implement a AA tree in Java:

// AATree class
//
// CONSTRUCTION: with no initializer
//
// ******************PUBLIC OPERATIONS*********************
// void insert( x )       --> Insert x
// void remove( x )       --> Remove x
// Comparable find( x )   --> Return item that matches x
// Comparable findMin( )  --> Return smallest item
// Comparable findMax( )  --> Return largest item
// boolean isEmpty( )     --> Return true if empty; else false
// void makeEmpty( )      --> Remove all items
// ******************ERRORS********************************
// Exceptions are thrown by insert and remove if warranted

/**
 * Implements an AA-tree.
 * Note that all "matching" is based on the compareTo method.
 * @author Mark Allen Weiss
 */
public class AATree 
{
    /**
     * Construct the tree.
     */
    public AATree( ) {
        root = nullNode;
    }
    
    /**
     * Insert into the tree.
     * @param x the item to insert.
     * @throws DuplicateItemException if x is already present.
     */
    public void insert( Comparable x ) {
        root = insert( x, root );
    }
    
    /**
     * Remove from the tree.
     * @param x the item to remove.
     * @throws ItemNotFoundException if x is not found.
     */
    public void remove( Comparable x ) {
        deletedNode = nullNode;
        root = remove( x, root );
    }
    
    /**
     * Find the smallest item in the tree.
     * @return the smallest item or null if empty.
     */
    public Comparable findMin( ) {
        if( isEmpty( ) )
            return null;
        
        AANode ptr = root;
        
        while( ptr.left != nullNode )
            ptr = ptr.left;
        
        return ptr.element;
    }
    
    /**
     * Find the largest item in the tree.
     * @return the largest item or null if empty.
     */
    public Comparable findMax( ) {
        if( isEmpty( ) )
            return null;
        
        AANode ptr = root;
        
        while( ptr.right != nullNode )
            ptr = ptr.right;
        
        return ptr.element;
    }
    
    /**
     * Find an item in the tree.
     * @param x the item to search for.
     * @return the matching item of null if not found.
     */
    public Comparable find( Comparable x ) {
        AANode current = root;
        nullNode.element = x;
        
        for( ; ; ) {
            if( x.compareTo( current.element ) < 0 )
                current = current.left;
            else if( x.compareTo( current.element ) > 0 )
                current = current.right;
            else if( current != nullNode )
                return current.element;
            else
                return null;
        }
    }
    
    /**
     * Make the tree logically empty.
     */
    public void makeEmpty( ) {
        root = nullNode;
    }
    
    /**
     * Test if the tree is logically empty.
     * @return true if empty, false otherwise.
     */
    public boolean isEmpty( ) {
        return root == nullNode;
    }
    
    /**
     * Internal method to insert into a subtree.
     * @param x the item to insert.
     * @param t the node that roots the tree.
     * @return the new root.
     * @throws DuplicateItemException if x is already present.
     */
    private AANode insert( Comparable x, AANode t ) {
        if( t == nullNode )
            t = new AANode( x );
        else if( x.compareTo( t.element ) < 0 )
            t.left = insert( x, t.left );
        else if( x.compareTo( t.element ) > 0 )
            t.right = insert( x, t.right );
        else
            throw new DuplicateItemException( x.toString( ) );
        
        t = skew( t );
        t = split( t );
        return t;
    }
    
    /**
     * Internal method to remove from a subtree.
     * @param x the item to remove.
     * @param t the node that roots the tree.
     * @return the new root.
     * @throws ItemNotFoundException if x is not found.
     */
    private AANode remove( Comparable x, AANode t ) {
        if( t != nullNode ) {
            // Step 1: Search down the tree and set lastNode and deletedNode
            lastNode = t;
            if( x.compareTo( t.element ) < 0 )
                t.left = remove( x, t.left );
            else {
                deletedNode = t;
                t.right = remove( x, t.right );
            }
            
            // Step 2: If at the bottom of the tree and
            //         x is present, we remove it
            if( t == lastNode ) {
                if( deletedNode == nullNode || x.compareTo( deletedNode.element ) != 0 )
                    throw new ItemNotFoundException( x.toString( ) );
                deletedNode.element = t.element;
                t = t.right;
            }
            
            // Step 3: Otherwise, we are not at the bottom; rebalance
            else
                if( t.left.level < t.level - 1 || t.right.level < t.level - 1 ) {
                if( t.right.level > --t.level )
                    t.right.level = t.level;
                t = skew( t );
                t.right = skew( t.right );
                t.right.right = skew( t.right.right );
                t = split( t );
                t.right = split( t.right );
                }
        }
        return t;
    }
    
    /**
     * Skew primitive for AA-trees.
     * @param t the node that roots the tree.
     * @return the new root after the rotation.
     */
    private static AANode skew( AANode t ) {
        if( t.left.level == t.level )
            t = rotateWithLeftChild( t );
        return t;
    }
    
    /**
     * Split primitive for AA-trees.
     * @param t the node that roots the tree.
     * @return the new root after the rotation.
     */
    private static AANode split( AANode t ) {
        if( t.right.right.level == t.level ) {
            t = rotateWithRightChild( t );
            t.level++;
        }
        return t;
    }
    
    /**
     * Rotate binary tree node with left child.
     */
    private static AANode rotateWithLeftChild( AANode k2 ) {
        AANode k1 = k2.left;
        k2.left = k1.right;
        k1.right = k2;
        return k1;
    }
    
    /**
     * Rotate binary tree node with right child.
     */
    private static AANode rotateWithRightChild( AANode k1 ) {
        AANode k2 = k1.right;
        k1.right = k2.left;
        k2.left = k1;
        return k2;
    }
    
    private static class AANode {
        // Constructors
        AANode( Comparable theElement ) {
            element = theElement;
            left    = right = nullNode;
            level   = 1;
        }
        
        Comparable element;      // The data in the node
        AANode     left;         // Left child
        AANode     right;        // Right child
        int        level;        // Level
    }
    
    private AANode root;
    private static AANode nullNode;
    
    static         // static initializer for nullNode
    {
        nullNode = new AANode( null );
        nullNode.left = nullNode.right = nullNode;
        nullNode.level = 0;
    }
    
    private static AANode deletedNode;
    private static AANode lastNode;
    
    // Test program; should print min and max and nothing else
    public static void main( String [ ] args ) {
        AATree t = new AATree( );
        final int NUMS = 40000;
        final int GAP  =   307;
        
        System.out.println( "Checking... (no bad output means success)" );
        
        t.insert( new Integer( NUMS * 2 ) );
        t.insert( new Integer( NUMS * 3 ) );
        for( int i = GAP; i != 0; i = ( i + GAP ) % NUMS )
            t.insert( new Integer( i ) );
        System.out.println( "Inserts complete" );
        
        t.remove( t.findMax( ) );
        for( int i = 1; i < NUMS; i+= 2 )
            t.remove( new Integer( i ) );
        t.remove( t.findMax( ) );
        System.out.println( "Removes complete" );
        
        
        if( ((Integer)(t.findMin( ))).intValue( ) != 2 ||
                ((Integer)(t.findMax( ))).intValue( ) != NUMS - 2 )
            System.out.println( "FindMin or FindMax error!" );
        
        for( int i = 2; i < NUMS; i+=2 )
            if( ((Integer)t.find( new Integer( i ) )).intValue( ) != i )
                System.out.println( "Error: find fails for " + i );
        
        for( int i = 1; i < NUMS; i+=2 )
            if( t.find( new Integer( i ) )  != null )
                System.out.println( "Error: Found deleted item " + i );
    }
}


/**
 * Exception class for duplicate item errors
 * in search tree insertions.
 * @author Mark Allen Weiss
 */
public class DuplicateItemException extends RuntimeException 
{
    /**
     * Construct this exception object.
     */
    public DuplicateItemException( ) {
        super( );
    }
    /**
     * Construct this exception object.
     * @param message the error message.
     */
    public DuplicateItemException( String message ) {
        super( message );
    }
}


/**
 * Exception class for failed finds/removes in search
 * trees, hash tables, and list and tree iterators.
 * @author Mark Allen Weiss
 */
public class ItemNotFoundException extends RuntimeException 
{
    /**
     * Construct this exception object.
     */
    public ItemNotFoundException( ) {
        super( );
    }
    
    /**
     * Construct this exception object.
     * @param message the error message.
     */
    public ItemNotFoundException( String message ) {
        super( message );
    }
}